a piece of wire 12 ft. long is cut into two pieces. one piece is made into a circle and the other piece is made into a square. Let the piece of length x be formed into a circle. allow x to equal 0 or 12, so all the wire is used for the square or for the circle. How long should each piece of wire be to minimize the total area? what is the radius of the circle? how long is each side of the square?
Let the radius of the circle be r and each side of the square be x
so 2πr + 4x = 12
πr + 2x = 6
x = (6-πr)/2
Area = πr^2 + x^2
= πr^2 + ((6-πr)/2)^2
= πr^2 + (3 - πr/2)^2
d(Area)/dr = 2πr + 2(3-πr/2) (-π/2)
= 2πr - π(3-πr/2)
= 0 for a minimus Area
2πr = π(3-πr/2)
2r = 3-πr/2
4r = 6 - πr
4r+πr = 6
r = 6/(4+π) = appr .840
so we need 2πr or 5.2788 ft for the circle,
leaving 6.721 ft for the square.
radius of circle = .840 ft
side of each square = 1.68 ft
check:
2πr + 4x
= 2π(.840) + 4(1.68) = 11.998 , not bad
let r = .8 , then x = 1.743
area = π(.8)^2 + 1.743^2 = 5.049
let r = .9, then x = 1.586
area = π(.9)^2 + (1.586)^2 = 5.06.
our answer
r = .84 , x = 1.68
area = π(.84)^ + 1.68^2 = 5.039 , which is lower than either of the
slightly larger and slightly smaller radii.
To minimize the total area, we need to find the values for x that minimize the sum of the areas of the circle and square.
Let's start by considering the case when x = 0. In this case, the entire wire is used for the square. The perimeter of the square is equal to the length of the wire, which is 12 ft. Since a square has all sides equal in length, each side of the square would be 12/4 = 3 ft. Therefore, the square would have an area of 3^2 = 9 sq. ft.
Next, let's consider the case when x = 12. In this case, the entire wire is used for the circle. The circumference of the circle is equal to the length of the wire, which is 12 ft. Therefore, the radius of the circle would be 12/(2π) = 12/6.28 ≈ 1.91 ft. The area of the circle can be calculated using the formula A = πr^2, where r is the radius. So, the area of the circle would be π(1.91)^2 ≈ 11.47 sq. ft.
To compare the two cases, we have:
Case 1: Square with side length of 3 ft, area = 9 sq. ft
Case 2: Circle with radius of 1.91 ft, area ≈ 11.47 sq. ft
Since the area of the circle is larger than the area of the square, the minimum total area will occur when all the wire is used for the circle. So, the length of the wire x that should be used to form the circle would be 12 ft.
Therefore:
- The radius of the circle is approximately 1.91 ft.
- The side length of the square is 3 ft.
To find the lengths that minimize the total area, we need to consider the formulas for the area of a circle and the area of a square.
1. Let's consider the piece of wire made into a circle first. We will use the formula for the circumference of a circle:
C = 2πr, where C is the circumference and r is the radius.
In this case, if x ft of wire is used for the circle, we have:
C = x, so 2πr = x, and solving for r, we get r = x / (2π).
Now, the area of a circle is given by the formula:
A = πr^2, where A is the area of the circle.
Substituting the value of r we found earlier, we get:
A = π(x / (2π))^2 = πx^2 / (4π^2) = x^2 / (4π).
2. Now, let's consider the piece of wire made into a square. The perimeter of a square is equal to four times the length of one side. So, if (12 - x) ft of wire is used for the square, we have:
4s = (12 - x), where s is the length of one side of the square.
Solving for s, we get s = (12 - x) / 4 = (3 - x/4).
The area of a square is simply the side length squared:
A = s^2 = [(3 - x/4)]^2 = (9 - 3x/2 + x^2/16).
3. To minimize the total area, we need to find the values of x that minimize the expression A_total = A_circle + A_square.
A_total = x^2 / (4π) + (9 - 3x/2 + x^2/16).
Taking the derivative of A_total with respect to x and setting it equal to zero, we can find the potential minimum:
d(A_total)/dx = 0.
Simplifying, we get: d(A_total)/dx = (8πx - 6 + x/8) / (4π) = 0.
Solving for x, we find x = 24 / (8π + 1).
4. To find the radius of the circle, we can substitute the value of x we found back into the equation for r:
r = x / (2π) = (24 / (8π + 1)) / (2π) = 12 / (4π + 1).
5. To find the length of each side of the square, we can substitute the value of x into our expression for s:
s = (3 - x/4) = (3 - (24 / (4π + 1))/4) = 3 - (6 / (4π + 1)).
Therefore, by using x = 24 / (8π + 1) ft of wire for the circle, we will minimize the total area. The radius of the circle will be 12 / (4π + 1) ft, and the length of each side of the square will be 3 - (6 / (4π + 1)) ft.